Heat the oil in a large stock pot or Dutch oven over medium. Add the onion and poblano and cook, stirring occasionally, until they have softened without taking on any color, 4 to 6 minutes. Add the garlic, cumin, coriander, oregano and red-pepper flakes, if using, and cook, stirring frequently, until the garlic and spices are fragrant, about 1 minute more. Season well with salt and pepper.
Step 2
Add the stock and chicken and bring to a simmer. Cook until the chicken has cooked through and is no longer pink, 10 to 12 minutes. Using tongs or a slotted spoon, transfer the cooked chicken to a cutting board. When cool enough to handle, shred the meat.
Step 3
Add the shredded chicken back to the pot along with the hominy and allow everything to simmer together for 5 to 10 minutes more. Season to taste with salt and pepper.
Step 4
Ladle into soup bowls. Serve with limes, for squeezing on top, and bowls of optional garnishes.